Biomedical Engineering Reference
In-Depth Information
or by Tresca criterion (mode II). As soon as the criterion is met, the crack propagates
to the next element boundary, additional enrichments are added to accommodate for
the extra length of the crack and the criterion is tested for the next element.
23.3 Numerical Description
The weak form for the finite element method is derived by standard Galerkin ap-
proach. Then the weak equations are discretized leading to a time-dependent, non-
linear system. This is solved using a Crank-Nicolson scheme for time-integration
and Newton-Raphson iteration within each time increment. A discretized form is
derived by dividing body
Ω
into elements
Ω
e
,
e
=
n
1
Ω
e
). The
result is that also the discontinuity is discretized in elements
S
d
and the boundary
in elements
S
e
. The displacements, the chemical potential and their variations are
discretized similarly (Bubnov-Galerkin approach) by
=
1
,...,n
e
(
Ω
~
T
~
u
,
~
T
~
u
,
μ
f
~
T
~
μ
,
μ
f
~
T
~
μ
,
u
ˆ
=
u
˜
=
ˆ
=
˜
=
(23.39)
where
~
=[
contains the shape functions. The columns
~
u
and
~
u
contain
the nodal values for bulk part and enhanced part, respectively. Similar are
~
,
~
μ
and
~
μ
columns of shape functions and nodal values. The introduction of fluid flow
does demands a time stepping algorithm. Time stepping here is therefore driven
by diffusion of the fluid and not dissipation of energy. The solution is sensitive to
the magnitude of the time increment. A large time step leads to underestimation
of fluid pressure in confined compression. Taking too small steps leads to initial
oscillation. For stable time integration the following law has to be satisfied (Vermeer
and Verruijt,
1981
)
~
x
~
y
]
t>
x
2
cK
,
(23.40)
in which
x
is characteristic size of an element and
t
is the time step. A Crank-
Nicolson scheme is used. Although the bulk material is assumed linear elastic, the
presence of damage introduces nonlinearity. The system is therefore solved itera-
tively at each time step. The matrices involved are given elsewhere (Kraaijeveld et
al.,
2009
). The model has been programmed using the Jem/Jive finite element toolkit
which has been developed by Habanera (Rijswijk, Netherlands). For implementa-
tion aspects like the tracking of the crack tip, increasing the degrees of freedom
or other propagation issues we refer to Remmers et al. (
2003
,
2008
) and Remmers
(
2006
).
